Sur les fonctions finement holomorphes
Annales de l'Institut Fourier, Volume 31 (1981) no. 4, p. 57-88
These functions are defined in open sets in the Brelot-Cartan fine topology in the complex plane. They generalize the ordinary holomorphic functions. The study of the finely holomorphic functions is bases here on the Beppo Levi functions, made precise in the sense of Deny. Using the Cauchy-Pompeiu transform the results by Debiard, Gaveau, and Lyons are recovered and extended in a non-probabilistic way. In addition it is shown that every finely holomorphic function is uniquely determined by its Taylor series, and that this series represents the function finely locally in an asymptotic sense. Furthermore, a finely holomorphic function has at most countably many zeros. At the end the finely holomorphic functions of class L p are studied.
Ces fonctions sont définies dans des ouverts pour la topologie fine de Brelot-Cartan dans le plan complexe. Elles généralisent les fonctions holomorphes ordinaires. L’étude des fonctions finement holomorphes est fondée ici sur les fonctions Beppo Levi comme précisées par Deny. En utilisant la transformée de Cauchy-Pompeiu on retrouve et étend de façon non-probabiliste les résultats de Debiard, Gaveau et Lyons. On montre en outre que toute fonction finement holomorphe est déterminée par sa série de Taylor. Cette série représente la fonction finement localement dans un sens asymptotique. Le nombre des zéros d’une fonction finement holomorphe est au plus dénombrable. À la fin on étudie les fonctions finement holomorphes de la classe L p .
@article{AIF_1981__31_4_57_0,
     author = {Fuglede, Bent},
     title = {Sur les fonctions finement holomorphes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {31},
     number = {4},
     year = {1981},
     pages = {57-88},
     doi = {10.5802/aif.849},
     mrnumber = {83e:31003},
     zbl = {0445.30040},
     language = {fr},
     url = {https://aif.centre-mersenne.org/item/AIF_1981__31_4_57_0}
}
Sur les fonctions finement holomorphes. Annales de l'Institut Fourier, Volume 31 (1981) no. 4, pp. 57-88. doi : 10.5802/aif.849. https://aif.centre-mersenne.org/item/AIF_1981__31_4_57_0/

[1] M. Brelot, Introduction axiomatique de l'effilement, Annali Mat. Pura Appl., (IV), 57 (1962), 77-96. | MR 25 #3187 | Zbl 0119.08902

[2] A. Debiard et B. Gaveau, Potentiel fin et algèbres de fonctions analytiques I, J. Funct. Anal., 16 (1974), 289-304. | MR 52 #1326 | Zbl 0297.31004

[3] A. Debiard et B. Gaveau, Potentiel fin et algèbres de fonctions analytiques II, J. Funct. Anal., 17 (1974), 296-310. | MR 380427 | MR 52 #1327 | Zbl 0301.31002

[4] J. Deny et J. L. Lions, Les espaces du type de Beppo Levi, Ann. Inst. Fourier, 5 (1953-1954), 305-370. | Numdam | MR 17,646a | Zbl 0065.09903

[5] B. Fuglede, Connexion en topologie fine et balayage des mesures, Ann. Inst. Fourier, 21, 3 (1971), 227-244. | Numdam | MR 49 #9241 | Zbl 0208.13802

[6] B. Fuglede, Finely Harmonic Functions, Springer Lecture Notes in Mathematics, 289, Berlin-Heidelberg-New York, 1972. | MR 56 #8883 | Zbl 0248.31010

[7] B. Fuglede, Fonctions harmoniques et fonctions finement harmoniques, Ann. Inst. Fourier, 24, 4 (1974), 77-91. | Numdam | MR 51 #3490 | Zbl 0287.31003

[8] B. Fuglede, Asymptotic paths for subharmonic functions, Math, Ann. 213 (1975) ; 261-274. | MR 52 #775 | Zbl 0283.31001

[9] B. Fuglede, Sur la fonction de Green pour un domaine fin, Ann. Inst. Fourier, 25, 3-4 (1975), 201-206. | Numdam | MR 55 #3289 | Zbl 0289.31012

[10] B. Fuglede, Finely harmonic mappings and finely holomorphic functions, Ann. Acad. Sci. Fennicæ (A.I.), 2 (1976), 113-127. | MR 57 #9998 | Zbl 0345.31008

[11] B. Fuglede, Invariant characterization of the fine topology in potential theory, Math. Ann., 241 (1979), 187-192. | MR 80k:31007 | Zbl 0402.31015

[12] B. Fuglede, Fine topology and finely holomorphic functions, To appear in the Proceedings of the 18th Scandinavian Congress of Mathematicians, Aarhus 1980. Birkhäuser Verlag.

[13] B. Fuglede, Fonctions BLD et fonctions finement surharmoniques, KUMI Preprint Series n° 15, Copenhagen 1980. (A paraître dans Séminaire de Théorie du Potentiel, Paris.) | Zbl 0484.31003

[14] V. P. Havin, Approximation in the mean by analytic functions, Dokl. Akad. Nauk SSSR, 1978, 5 (1968). (Russian)) — Soviet Math. Dokl. 9, 1 (1968), 245-248. | Zbl 0182.40201

[15] T. Lyons, Finely holomorphic functions, J. Funct. Anal., 37 (1980), 1-18. | MR 82d:31011a | Zbl 0459.46038

[16] T. Lyons, A theorem in fine potential theory and applications to finely holomorphic functions, J. Funct. Anal., 37 (1980), 19-26. | MR 82d:31011b | Zbl 0459.46039

[17] T. Lyons and A. G. O'Farrell. (Personal communication.)

[18] R. Mckissic, A non-trivial normal sup norm algebra, Bull. Amer. Math. Soc., 69 (1963), 391-395. | Zbl 0113.31502

[19] Nguyen-Xuan-Loc, Sur la théorie des fonctions finement holomorphes. Bull. Sci. Math., 102 (1978), 271-308. | MR 80c:31006 | Zbl 0389.60060

[20] Nguyen-Xuan-Loc, Sur la théorie des fonctions finement holomorphes (II). Dans : Complex Analysis Joensuu, 1978, 289-300. Springer Lecture Notes in Mathematics, 747, Berlin-Heidelberg-New York, 1979. | Zbl 0435.60079

[21] Nguyen-Xuan-Loc, Singularities of locally analytic processes (with applications in the study of finely holomorphic and finely harmonic functions). In : Potential Theory, Copenhagen, 1979, 267-288. Springer Lecture Notes in Mathematics, 787. Berlin-Heidelberg-New York, 1980. | Zbl 0478.60081

[22] G. Polya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, n° 27, Princeton, 1951. | MR 13,270d | Zbl 0044.38301

[23] I. N. Vekua, Generalized Analytic Functions, Oxford-London-New York-Paris, 1962. | Zbl 0100.07603

[24] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89. | JFM 60.0217.01 | MR 1501735 | Zbl 0008.24902

[25] B. Øksendal, Finely harmonic functions need not be differentiable quasi-everywhere, (Manuscript, 1980).